In mathematics, a Cauchy matrix, named after Augustin-Louis Cauchy, is an m×n matrix with elements a_ij in the form

where ${\displaystyle x_{i}}$ and ${\displaystyle y_{j}}$ are elements of a field ${\displaystyle {\mathcal {F}}}$, and ${\displaystyle (x_{i})}$ and ${\displaystyle (y_{j})}$ are injective sequences (they contain distinct elements).

Every submatrix of a Cauchy matrix is itself a Cauchy matrix.

The Hilbert matrix is a special case of the Cauchy matrix, where

The determinant of a Cauchy matrix is clearly a rational fraction in the parameters ${\displaystyle (x_{i})}$ and ${\displaystyle (y_{j})}$. If the sequences were not injective, the determinant would vanish, and tends to infinity if some ${\displaystyle x_{i}}$ tends to ${\displaystyle y_{j}}$. A subset of its zeros and poles are thus known. The fact is that there are no more zeros and poles:

The determinant of a square Cauchy matrix A is known as a Cauchy determinant and can be given explicitly as

It is always nonzero, and thus all square Cauchy matrices are invertible. The inverse A⁻¹ = B = [b_ij] is given by

where A_i(x) and B_i(x) are the Lagrange polynomials for ${\displaystyle (x_{i})}$ and ${\displaystyle (y_{j})}$, respectively. That is,